A. E. CONKADY' 61 



aberration arising at each surface in the absolute measure of wave- 

 lengths, it also enables a designer to avoid the unnecessary piling up 

 of huge aberrations such as are met with in the lens systems designed 

 by purely geometrical ray-tracing. 



Recently (Monthly Notices, June, 1919), I have rounded off this 

 earlier work by determining the complete light-distribution in the 

 " spurious disc " which results when residuals of aberration are 

 present, so that the designer using the optical path method can now 

 state definitely to what extent the image points obtained with a 

 given system fall short of the full brightness and sharpness which 

 would result in a theoretically perfect instrument. 



The chromatic aberration of microscopic objectives is also best and 

 most conveniently determined in terms of differences of optical paths 

 (Monthly Notices, January and March, 1904). By applying the 

 simple formulae to both marginal and paraxial rays, a reliable 

 measure of the higher chromatic aberrations, the so-called spherical 

 variation of chromatic correction, is obtained, and this can then, 

 by suitable alterations of lens curvatures, etc., be kept within those 

 narrow limits which distinguish " semi-apochromatic " objectives 

 from earlier types in which this variation frequently reached very 

 serious amounts. 



A microscope objective perfectly free from spherical and chromatic 

 aberration may yet be absolutely useless for practical purposes on 

 account of such amounts of coma in the images of extra-axial object 

 points that sharp definition is limited to an almost infinitesimally 

 small area in the exact centre of the field. One of Abbe's first 

 attempts at the designing of microscope objectives purely by calcula- 

 tion appears to have resulted in a particularly bad specimen of this 

 type. The search for the cause of the defect led him to the inde- 

 pendent discovery of the famous " Sine-Condition," also announced 

 almost simultaneously by Helmholtz, and previously discovered — 

 without attracting the attention of opticians — by Clausius. In an 

 approximate algebraical form it also figured as the second of the 

 famous 5 conditions of Seidel. The realisation of its immense value, 

 however, dates undoubtedly from the announcements by Abbe and 

 Helmholtz in 1873. Since that time it has saved an incalculable 

 amount of time and trouble to the designers of telescope and micro- 

 scope objectives, as it indicates the presence or absence of coma in 

 the central part of the field by a simple comparison of figures taken 

 directly from the trigonometrical computations. I gave a simple 

 and fairly exhaustive proof and discussion of this theorem in Monthly 

 Notices for March, 1905, and to that paper those interested may 

 refer. 



If, and only if, the foregoing defects (spherical and chromatic 

 aberration within the Rayleigh limit and coma) are properly cor- 

 rected, then another defect of all ordinary lens systems will become 

 obvious and objectionable, viz., the secondary spectrum. This is 

 duB to the fact that, as compared with ordinaiy crown glasses, the 

 heavy flint glasses which have to be used to compensate the primary 

 chromatic aberration disperse the blue end of the spectrum too much 

 and the red end too little. The result is that flint lenses of the 

 proper power to secure achromatism for the brightest yellow and 

 green region of the spectrum overcorrect the dispersion of the crown 



